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Abstract
Graded versions of the principal series modules of the category O of a semisimple complex Lie algebra g are defined. Their combinatorial descriptions are given by some Kazhdan–Lusztig polynomials. A graded version of the Duflo–Zhelobenko four-term exact sequence is proved. This gives results about composition factors of quotients of the universal enveloping algebra of g by primitive ideals; in particular an upper bound is obtained for the multiplicities of such composition factors. Explicit descriptions are given of principal series modules for Lie algebras of rank 2. It can be seen that these graded versions of principal series representations are neither rigid nor Koszul modules.
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